Stochastic ProcessesDr. Talal Skaik

Chapter 10

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Probability and Stochastic ProcessesA friendly introduction for electrical and computer engineers

Electrical Engineering departmentIslamic University of Gaza

December 2011

•The word stochastic means random. •The word process in this context means function of time.

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Example: where is a uniformly distributed random variable in represents a stochastic process.

),cos()( 0 tatX (0,2 ),

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Ensemble average:With t fixed at t=t0, X(t0) is a random variable, we have the averages ( expected value and variance) as we studied earlier.Time average: applies to a specific sample function x(t, s0), and produces a typical number for this sample function.

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For a specific t, X(t) is a random variable with distribution:

])([),( xtXptxF xtxFtxf

),(),(

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When Cov[X,Y] is applied to two random variables that are observations of X(t) taken at two different times, t1 and t2

=t1 +τ seconds:The covariance indicates how much the process is likely to change in the τ seconds elapsed between t1 and t2. A high covariance indicates that the sample function is unlikely to change much in the τ-second interval.A covariance near zero suggests rapid change.

Autocovariance

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Recall in a stochastic process X(t), there is a random variable X(t1) at every time t1 with PDF fX(t1)(x).For most random processes, the PDF fX(t1)(x) depends on t1. For a special class of random processes know as stationary processes, fX(t1)(x) does not depend on t1.Therefore: the statistical properties of the stationary process do not change with time (time-invariant).

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